Shocks
For more details see for example Goedbloed, Keppens & Poedts, 2010 Advanced Magnetohydrodynamics.Notations and definitions
The velocity is decomposed \begin{equation} \label{eq:ms2} \vec{u} = u_{n}\vec{e}_{n} + u_{t} \vec{e}_{t} \end{equation} in a component \(u_{n}\) parallel to the vector normal to the shock (the normal vector \(\vec{n}\)), and a tangential component \(\vec{v}_{t}\). Note, that there are two tangential directions perpendicular to each other. The indices \(n,t\) are then the normal, respective tangenial components of the vectors, \(\rho,P, \vec{u},\vec{B}, \gamma\) are the denisty, thermal pressure, plasma (bulk) velocity, magnetic field and polytropic index. The indes \(i\in\{1,2\}\) denote the upstream, repsective the downstream direction. Further, we will use \(\vec{B} = \vec{B}' / \sqrt{\mu}\), \(\vec{B}'\) is the magnetic induction in physical units (SI or cgs). The following notations will be used (\(i \in \{u,d\}\), where "u" are the parameter in upwind direction and "d" those downwind:| \(v_{A,n,i}\) | \(\equiv\) | \(\frac{B_{n,i}}{\sqrt{\rho}}\) | normal Alvfén speed |
| \(v_{c}\) | \(\equiv\) | \( \sqrt{\frac{\gamma P}{\rho}}\) | sound speed |
| \(M_{A,n,i}\) | \(\equiv\) | \(\frac{u_{n,i}}{V_{A,n,i}}\) | normal Alvfénic Mach number |
| \(M\) | \(\equiv\) | \(M_{A,n,u}\) | shorthand notation |
| \(M_{A,t,i}\) | \(\equiv\) | \(\frac{B_{t,i}}{u_{n,i}\sqrt{\rho_{i}}}\) | ``tangential'' Alfv\'enic Mach number |
| \(M_{s,n,i}\) | \(\equiv\) | \(\frac{u_{n}}{v_{c}}\) | normal sonic Mach number |
| \(s\) | \(\equiv\) | \(\frac{\rho_{d}}{\rho_{u}}=\frac{u_{n,u}}{u_{n,d}}\) | the compression ratio |
| \(\beta_{t,i} \) | \(\equiv\) | \(2P_{i} /B^{2}_{t,i}\) | tangential plasma beta |
| \(=\frac{2}{\gamma}\frac{\gamma \rho_{i}u_{n,i}^{2}P_{i}}{\rho_{i}u_{n,i}^{2}B^{2}_{t,i}}\) | \(=\frac{2}{\gamma}\, \frac{v_{c,i}^{2}}{v_{A,t,i}^{2}} =\frac{2}{\gamma}\, \frac{M_{A,t,i}}{M_{s,n,i}}\) | alternative represnetation | |
| \(\beta_{n,i} \) | \(\equiv\) | \(2P_{i} /B^{2}_{n,i}\) | normal plasma beta |
| \(=\frac{2}{\gamma}\frac{\gamma \rho_{i}u_{n,i}^{2}P_{i}}{\rho_{i}u_{n,i}^{2}B^{2}_{n,i}}\) | \(=\frac{2}{\gamma}\, \frac{v_{c,i}^{2}}{v_{A,n,i}^{2}} =\frac{2}{\gamma}\, \frac{M_{A,n,i}}{M_{s,n,i}}\) | alternative represnetation | |
| \(\beta_{i}\) | \(\equiv\) | \(\beta_{n,i}\) | shorthand notation |
We define the following angles:
\begin{align}
\vartheta_{i} &= \measuredangle \vec{B}_{i},\vec{n} \to \tan\vartheta_{i}=\frac{B_{t,i}}{B_{n}}\\
\varphi_{i} &= \measuredangle \vec{u}_{i},\vec{n} \to \tan\varphi_{i}=\frac{u_{t,i}}{u_{n}}\\
\alpha_{i} &= \measuredangle \vec{B}_{i},\vec{u}_{i}\to \cos\alpha_{i}
=\frac{\vec{u}_{i}\cdot \vec{B}_{i}}{u_{i}B_{i}}
\end{align}
To calculate the Rankine Hugoniot equations, we replace \(\partial d/\partial t \to -U\)
where \(U\) is the shock speed and \(\vec{\nabla}\to \vec{n}\) and switch to
the shock rest frame \(\vec{u}' = \vec{u} - \vec{U}\). To save writings we negfelct the prime at \(\vec{u}' \) in the following.
Further, we will use \(\vec{B} = \vec{B}' /\sqrt{\mu}\), \(\vec{B}'\) is the magnetic induction in physical units
(SI or cgs).
The Rankine Hugoniot equations
Then we obtain for the Rankine Hugoniot equations in the shock rest frame: \begin{align} \left\{ \rho u_{n}\right\} &=0 &\text{(continuity)}&\\ \left\{\rho u_{n}^{2} +P + \frac{1}{2} B^{2}_{t}\right\} &=0 &\text{(normal momentum)} &\\ \rho u_{n}\left\{\vec{u}_{t}\right\}- B_{n}\left\{\vec{B}_{t}\right\} &=0 &\text{(tangential momentum)} &\\ \rho u_{n}\left\{\frac{1}{2} (u^{2}_{t}+u_{n}^{2}) + \frac{1}{\rho}\left(\frac{\gamma}{\gamma-1} P+\frac{B^{2}_{t}}{2}\right)\right\} -B_{n}\left\{(\vec{u}_{t}\cdot\vec{B}_{t})\right\} &=0 &\text{(energy)}&\\ \left\{B_{n}\right\} &=0 &\text{(normal B-flux)} &\\ \rho u_{n} \left\{\frac{\vec{B}_{t}}{\rho}\right\}-B_{n}\left\{\vec{u}_{t}\right\}&= 0 &\text{(tangential B-flux)} &\\ \end{align} where the curly brackets are the shorthand notation for \(\{a\}=a_u-a_d\) where the indices \(u,d\) are the upstream respctivly the downstream direction.
The above system are 8 equations for eight unkowns \(\rho, u_n, P, B_n\) and two vector components for \(\vec{u}_t\) and \(\vec{B}_t\). The shocks and discontinuities are characterized as follows:- hydrodynmic shock: \( B_n=0, \vec{B}_t=\vec{0} \)
- perpendicular shock: \( B_n = 0\)
- parallel shock: \( \vec{B}_t=\vec{0} \)
- slow shock: \(M_{A,n,2}^2\le M_{A,n,1}^2\le 1 \)
- intermediate shock: \(M_{A,n,2}^2\le 1\le M_{A,n,1}^2 \)
- fast shock_\( 1\le M_{A,n,2}^2\le M_{A,n,1}^2 \)
- switch on shock: \(B_{t,1} =0, B_{t,2}\ne 0\)
- switch off shock: \( B_{t,1} \ne 0, B_{t,2}= 0 \)
- contact disconinity: \(u_n=0, B_n\ne 0\Rightarrow \{\vec{u}_t\} =0, \{\vec{B}_t\} =0, \{P\}=0 \) but \(\rho\ne 0\)
- tangential discontuity: \(u_n=0, B_n= 0 \Rightarrow \left\{P+\frac{1}{2} B_t^2\right\} =0 \), but \( \{\vec{u}_t\} \ne0, \{\vec{B}_t\} \ne0, \{P\}\ne 0 \) ,\(\rho\ne 0\)
To caluclate the donwstrema values we have to solve the above equations system with respect t \(s\). The general case is a tedious algebrac excercise, where one has to handle the perpendicular shocks separatly, because, in general, because in the general case a divison by \(B_n\) is required. its solutions are found here.
